| Summary: | This thesis generalized the concept of stop-loss transforms, an important concept in risk theory (6), to the nth stop-loss transforms. Some useful properties of the nth stop-loss transforms were discovered and a recursion formula for the nth stop-loss transforms was established. Also, the maintenance properties of the nth stop-loss order under convolution, compound and mixture operations were proved. Finally the results mentioned above were applied to the study of losses $L\sb{i}\ (i = 1,2,\cdots),$ maximal aggregate loss L and ruin probability $T\psi(u).$ Some inequalities for the expectation of $L\sb{i}$ and L were given, and a relationship between the claim amount random variable and ruin probability was found.
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